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Simplifying 9(t) = -16t2 + 24t + 5 Reorder the terms: 9t = 5 + 24t + -16t2 Solving 9t = 5 + 24t + -16t2 Solving for variable 't'. Reorder the terms: -5 + 9t + -24t + 16t2 = 5 + 24t + -16t2 + -5 + -24t + 16t2 Combine like terms: 9t + -24t = -15t -5 + -15t + 16t2 = 5 + 24t + -16t2 + -5 + -24t + 16t2 Reorder the terms: -5 + -15t + 16t2 = 5 + -5 + 24t + -24t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -15t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 -5 + -15t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -5 + -15t + 16t2 = 0 + -16t2 + 16t2 -5 + -15t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -15t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -0.9375t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -0.9375t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -0.9375t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -0.9375t + t2 = 0 + 0.3125 -0.9375t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -0.9375t + t2 = 0.3125 The t term is -0.9375t. Take half its coefficient (-0.46875). Square it (0.2197265625) and add it to both sides. Add '0.2197265625' to each side of the equation. -0.9375t + 0.2197265625 + t2 = 0.3125 + 0.2197265625 Reorder the terms: 0.2197265625 + -0.9375t + t2 = 0.3125 + 0.2197265625 Combine like terms: 0.3125 + 0.2197265625 = 0.5322265625 0.2197265625 + -0.9375t + t2 = 0.5322265625 Factor a perfect square on the left side: (t + -0.46875)(t + -0.46875) = 0.5322265625 Calculate the square root of the right side: 0.729538596 Break this problem into two subproblems by setting (t + -0.46875) equal to 0.729538596 and -0.729538596.Subproblem 1
t + -0.46875 = 0.729538596 Simplifying t + -0.46875 = 0.729538596 Reorder the terms: -0.46875 + t = 0.729538596 Solving -0.46875 + t = 0.729538596 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.46875' to each side of the equation. -0.46875 + 0.46875 + t = 0.729538596 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + t = 0.729538596 + 0.46875 t = 0.729538596 + 0.46875 Combine like terms: 0.729538596 + 0.46875 = 1.198288596 t = 1.198288596 Simplifying t = 1.198288596Subproblem 2
t + -0.46875 = -0.729538596 Simplifying t + -0.46875 = -0.729538596 Reorder the terms: -0.46875 + t = -0.729538596 Solving -0.46875 + t = -0.729538596 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.46875' to each side of the equation. -0.46875 + 0.46875 + t = -0.729538596 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + t = -0.729538596 + 0.46875 t = -0.729538596 + 0.46875 Combine like terms: -0.729538596 + 0.46875 = -0.260788596 t = -0.260788596 Simplifying t = -0.260788596Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.198288596, -0.260788596}
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